Cross validation for a biased estimator of a Gaussian mean

Consider the mean estimator $$\hat{\mu}(\lambda) = \lambda \frac{1}{n}\sum_{i = 1}^nY_i$$ (for $n$ iid Gaussian variates $Y_i$). After calculating the bias and the variance of this estimator, I found the optimal value which minimizes the prediction error $$\text{Err}_\lambda = E(\mu - \hat{\mu}(\lambda) )^2= (\lambda-1)^2\mu^2+\lambda^2 \frac{\sigma^2}{n}$$ to be $$\lambda^*=\frac{\mu^2}{\frac{\sigma^2}{n}+\mu^2}.$$ Can I find this parameter by cross validation (k fold) and if so, how can I do that? For example, if I simulate a Gaussian with $\mu=0.1$, $\sigma = 0.2$, and $n = 30$, I should find $\lambda^*= 0.88$ with cross validation.

• An interesting aspect of this approach is that in finding $\lambda^*$ based on the data, it becomes a function of the data themselves--a statistic--rather than a constant, whence $\text{Err}_\lambda$ no longer has the same form as your formula, implying $\lambda^*$ is no longer necessarily optimal! – whuber Nov 2 '11 at 19:19
• i am agree but method like ridge or lasso for regression to ? – user7114 Nov 2 '11 at 20:05
• moreover if i consider i have an unbiased estimator of both $\mu$ and $\sigma$, i have an unbiased estimator of the error and the optimal parameter ? – grant Nov 3 '11 at 16:10
• Whuber do you have any idea about ? thanks – grant Nov 7 '11 at 13:44